A sphere is inscribed in a cube with edge length 9 inches. Then a smaller cube is inscribed in the sphere. How many cubic inches are in the volume of the inscribed cube? Express your answer in simplest radical form.
We draw a diagram:

[asy]
size(140);
draw(Circle((6,6),4.5));
draw((10.5,6)..(6,6.9)..(1.5,6),linetype("2 4"));
draw((10.5,6)..(6,5.1)..(1.5,6));
dot((6,6));
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle);
draw((0,9)--(3,12)--(12,12)--(9,9));
draw((12,12)--(12,3)--(9,0));
draw((0,0)--(3,3)--(12,3),dashed); draw((3,3)--(3,12),dashed);
[/asy]

The sphere's diameter length is equal to the big cube's side length, which is 9.

[asy]
size(100);
draw(Circle((6,6),9));
draw((15,6)..(6,8)..(-3,6),linetype("2 4"));
draw((15,6)..(6,4)..(-3,6));
dot((6,6));
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle);
draw((0,9)--(3,12)--(12,12)--(9,9));
draw((12,12)--(12,3)--(9,0));
draw((0,0)--(3,3)--(12,3),dashed); draw((3,3)--(3,12),dashed);
[/asy]

Now the sphere's diameter is equal to the space diagonal of the small cube, meaning that the distance between two opposite corners of a cube is equal to the diameter of the sphere.  To compute the space diagonal of the cube, let the side length of the cube be $s$, and label points $A$, $B$, $C$, $D$, $E$ as shown below.

[asy]
size(85);
pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps);
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle);
draw((0,9)--(3,12)--(12,12)--(9,9));
draw((12,12)--(12,3)--(9,0));
draw((0,0)--(3,3)--(12,3),dashed); draw((3,3)--(3,12),dashed);
label("$B$",(0,0),SW); label("$C$",(9,0),SE); label("$D$",(12,3),NE); label("$A$",(3,3),NW); label("$E$",(12,12),E);
[/asy] We look at triangle $\triangle BDE$, where $\overline{BE}$ is the space diagonal.  $\overline{DE}$ is a side length of the cube with length $s$. $\overline{BD}$ is the hypotenuse of an isosceles right triangle with legs length $s$, so its length is $\sqrt{s^2+s^2}=s\sqrt{2}$.  So we have \[BE=\sqrt{DE^2+BD^2}=\sqrt{s^2+(s\sqrt{2})^2} = \sqrt{3s^2} = s\sqrt{3}.\]Thus, the space diagonal of a cube with side length $s$ has length $s\sqrt{3}$.  The sphere has diameter 9, which is equal to the space diagonal of the cube, so we have \[9 = s\sqrt{3} \quad\Rightarrow \quad s = \frac{9}{\sqrt{3}}.\]Finally, the volume of the cube is $s^3 = \left(\frac{9}{\sqrt{3}}\right)^3 = \boxed{81\sqrt{3}}$.